MEASUREMENT: PROBLEMS OF THEORY AND APPLICATION

by

Patrick Suppes

TECHNICAL REPpRT NO. 147

June 12, 1969

PSYCHOLOGY SERIES

Reproduction in Whole or in Part is Permitted for

any Purpose of the United States Government

INSTITUTE FOR MATHEMATICAL STUDIES IN THE SOCIAL SCIENCES

STANFORD UNIVERSITY

STANFORD, CALIFORNIA

Measurement: Problems of Theory and Applicationl

Patrick Suppes

Institute for Mathematical Studies in the Social SciencesStanford University

Those of you who have just heard and enjoyed the fascinating lecture

by Professor Blakers may have a certain deja ~ sense as you hear what I

am going to say. This will be true especially of those who also attended

Mr. Mitchell's lecture earlier. To reduce this deja vu feeling, I shall

deal initially with ~uestions of measurement that have systematic or

formal interest, but that diverge rather sharply from the mainstream of

mathematics. From the standpoint of applying the theory of measurement

in the social sciences, or indeed even in the physical sciences, many

conceptually unsatisfactory things arise about a theory of measurement

that is related directly to standard structures in mathematics. The

structures widely studied in mathematics are almost without exception

infinitistic, and they are almost without exception error-free. In

concentrating on the problems of finitude and error, I shall organize

what I have to say under four headings.

First, I want to formulate a general viewpoint that is much in

agreement with that expressed by Professor Blakers. Second, I shall

discuss necessary and sufficient conditions for the existence of numerical

measures on finite structures. Third, I shall talk about algebraic theory

of error, and finally, I shall comment on nonalgebraic··error theory, with

some remarks about linear regression and the structural models discussed

by Professor Williams.

lThis paper is based on a lecture given at a UNESCO Seminar at theUniversity of Syndey, Australia on May 24, 1968. The content of the paperreflects research that has been supported by the National Science Foundation,

1

1, General VieyPoint

We begin with measurement procedures that are fundamental in the

sense that they assume no prior numerical results, We may represent the

empirical situation in most cases by a set A that is the set of objects

Or phenomena under consideration, and a finite sequence of finitary rela~:'

tions R. on this set, Such an object,l

ordinarily called a relational structure, I deliberately have chosen

relations rather than operations to represent the experimental or measure-

ment procedures of a fundamental character, because when we consider actual

procedures, the closure properties sO characteristic of ordinary mathematical

operations clearly lead us into infinistic idealizations that are not happy

idealizations for many applications of measurement, For example, if we

impose a closure condition on our algebra of operations far the measurement

of length, then we are committed almost at once to postulating the exis-

tence of lengths of arbitrarily great size, If we use relations instead

of operations, no such commitment is required, and we can restrict con-

siderations entirely to finite sets and finite relational structures,

(B,v a finite relational structure, I mean a relational structure in which

the basic set A of objects is finite,)

In this general viewpoint, two formal problems should be solved for

any fundamental measurement procedure or fundamental theory of measurement,

From a formal or mathematical standpoint, we characterize the class of

relational structures that satisfy the empirical procedure or the theory

by stating the axioms that must be satisfied by each structure, The first

formal problem then is to prove a representation theorem for any structure

satisfying the axioms, Ordinarily in order to call the theory a theory of

2

measurement, this representation theorem should show that any structure

. satisfying the axioms may be mapped homomorphically into the real numbers.

The restriction to the real numbers is not critical. Mapping into a

structure closely related to the real numbera is. For example, in the

case of multidimensional scaling, what is desired is a mapping into an

n-dimensional Euclidean space rather than the set of real numbers them­

selves. One point about the representation theorem needs clarification.

When the structures are finite there is no problem of mapping them homo­

morphically' into the real numbers with some arbitrary relations on the

real numbers. Tne interest of the problem is rather to provide in advance

the numerical relations in terms of which the numerical structures should

be defined. The homomorphism then must be relative to these given numerical

relations.

The second formal problem is that of uniqueness. How unique is

the homomorphism mapping a given structure into the real numbers? In

the classical measurement theory of mass or distance, for example, we

expect the mapping to be unique up to a positive similarity transformation.

This way of looking at theories of measurement is not special in any

sense to the domain of measurement procedures. In all areas of mathematics,

it is standard to search for representation theorems for structures of

primary interest and also to ask about the uniqueness of the representations

obtained. Professor Blakers has already mentioned the familiar classical

example, namely, the representation theorem for plane geometry in terms of

Cartesian coordinates, and the proof that this representation is unique

up to the group of Euclidean motions. In the context of the present

discussion, it is perhaps worth mentioning that classical synthetic

3

geometry can be axiomatized in an elementary fashion as a relational

structure ~ The baste set A j.B the set of poj_nts, and the two rela­

tions we define on the set A are the ternary relation of betweenness

between points and the quaternary relation of equidistance. The rela­

tion of equidistance means that the distance between points x and y

is the same as the distance between points u and v.

Still another example of considerable physical interest is to be

found in the relation between measurement and the theory of special

rela'dvity. In this case, greater interest has been attached to the

uniqueness theorem than to the representation theorem. We may show,

for example, that preservation of the relativistic length of inertial

segments is the only assumption needed to prove that any two inertial

frames are related by Lorentz transformations (Suppes 1959). More

recently, Zeeman (1964) has shown that by using slightly stronger assump­

tions about the number of dimensions (n ~ 3): it is possible to pos­

tulate that the time-like partial ordering of points is preserved in

order to obtain the Lorentz transformations. From the standpo:i.nt of

the theory of measurement, it is interesting to find that no additional

physics is required to derive the Lorentz transformations.

I mention these examples of geometry and relativity, because there

has been a recent tendency for the literature on the theory of measure­

ment to become isolated from other domains of sc:i.ence, and I regard this

as unfortunate in view of the close connect:i.ons of the sort just descr:i.bed.

2. Necessary and Sufficient Conditions

If we undertake to formulate a fundamental theory of measurement, .

we should recogn:i.ze that it is :i.mportant and interesting to state axioms

4

numerical structure

that are not only sufficient, but also necessary. Such axioms give us

a better sense of understanding the nature of the theory. There also' .are

more practical reasons of application for seeking necessary and sufficient

conditions. In considering collections of objects of a similar sort whose

properties are to be measured, we do not want to restrict the collections

of objects because of extraneous existential assumptions. We naturally

ask what minimal conditions can we assume guarantee the existence of a

measure? Existential conditions, for example, that are sufficient but

not necessary unduly restrict the range of structures that fall within

the theory. To seek necessary and sufficient conditions is to seek a

bare-bones characterization of all the structures that intuitively should

be brought within the framework of the theory.

At this point it may be useful to look at a simple example. Consider

binary structures, that is, structures consisting of a set A and a binary

relation R on this set. It is natural to ask what necessary and suf-

ficient conditions exist that make a binary structure homomorphic to a

n ~ < N, < > where N is a set of real numbers

and < is the usual numerical relation less than restricted to N.

If the set A is finite, the answer is simple. The relation R

must just be asymmetric, transitive, and connected on A. That is, the

following three axioms must be satisfied in the structure. For every

x, y, and z in A:

Axiom 1. If xRy then not yRX.--- ----Axiom 2. If xRy and yRz then xRz.---Axiom ). If x f y then xRy or yRx.

5

In the finite case, the proof of the necessity and the sufficiency is

obvious and need not be discussed further.

If we relax the restriction that the set A be finite, then the

three axioms are no longer sufficient, but only necessary. To see that

the axioms are not sufficient, we observe that they have models of arbi­

trary high cardinality. When a relation R satisfies these three axioms,

however, any homomorphism also must be an isomorphism, but there cannot

be a one-one function imbedding models of arbitrarily large infinite

cardinality into the real numbers. To imbed such an ordering in the

real numbers, we must add an additional condition. In the present case

a relatively simple answer is at hand. An ordering thiit;s'atfu~fies the ,above

three axioms also must have a countable order-dense subset in case the

set A is infinite. However, this condition for the infinite case is

not really interesting from the standpoint of the theory of measurement.

Because the necessary and sufficient conditions are so obvious and

simple for finite orderings, we initially might expect the situation to

be the same or close to the same for relational structures that seem

only slightly more complex than orderings. The next simple class to

consider is that of orderings on differences as well as orderings on

the objects themselves. For example, in a psychological experiment,

subjects might be asked to judge whether tone x is closer in pitch

to y than tone u is to v. In other words, we ask the subject to

make judgments about differences as well as order. More generally, we

can think of asking for judgments of relative similarity, that is, the

judgment that x is at least as similar to y as u is to v. The.

real-valued mapping f we want for such a quaternary relation D is this:

6

(1) xyDuv if and only if f(x) - f(y) ~ f(u) - f(v) 0

'Keeping this representation in mind, we shall call a relational structure

consisting of a set A and a quaternary relation D that satisfies this

condition a difference structure. The problem is to find the elementary

necessary and sufficient conditions that a difference structure must

satisfy to guarantee the existence of such a measurement function fo

As in the case of orderings, let us restrict ourselves to finite

sets and ask what sort of necessary and sufficient conditions we would

like to findo The simple thing about the three axioms on orderings is

that we can check a binary structure to see whether it satisfies the

axioms by looking at no more than triples of objectso To check on

asymmetry and connectedness, only pairs of objects need be examinedo

To check on transitivity, triples of objects need inspection to determine

whether there are any intransitive triadso

We would expect the situation to be somewhat more complex for

difference structures, but it still would be valuable as a first step

to seek generalizations of transitivityo For example, transitivity

of difference~whichwould require six variables for expression,would

demand a check on sextuples of objects in the set A to see if the

necessary and sufficient conditions for the existence of a measure

could be stated in a relatively simple way, especially in a way that

could be checked either by hand or with a simple computer programo It

would be desirable to have an upper bound on the size of the n-tuple

needed to check on the existence of a measure o An upper bound independent

of the cardinality of A shows that the structures for which a measure'

exists do not get essentially more complicated as the cardinality of the

7

set A increases. If we eliminate as axioms existential assertions,

then Dana Scott and I showed some years ago (1958) that for open sen­

tences or universal sentences, that is, for sentences that use only

universal ~uantifiers standing in front, it is not possible to give a

finite list of necessary and sufficient conditions that any finite structure

must satisfy in order to be a difference structure. In a subse~uent paper,

Scott (1964) gave necessary and sufficient conditions in terms of an in­

finite schema that increases in complexity as the number of objects in

the set A increases and that in terms of elementary open sentences is

e~uivalent to a countable infinity of such sentences. What is especially

important about this condition is that it re~uires checks on arbitrarily

large n-tuples of the number of objects when the set A increases.

To those interested in multidimensional scaling, it is worth re­

marking that the results Scott and I obtained have been generalized

recently to n dimensions by Titiev (1969), He shows that if we attempt

to represent similarity judgments in an n-dimensional space with an

Euclidean metric, then again necessary and sufficient conditions in

terms of a finite list of open sentences cannot be given. Titiev also

establishes a similar result for additive conjoint measurement in

n dimensionso

Perhaps the best necessary and sufficient conditions yet found in

a single paper are in Scott (1964), but his examination of other cases,

including measurement of subjective probabilities or measurements of mass

or distance, shows that a simple finite list of necessary and sufficient

conditions cannot be found in any of·the standard cases. In fact, it is

an interesting problem to ask for what cases other than those of simple

8

ordering can necessary and sufficient conditions be found for finite

domains? A closely related example is given in the next section, but

I must confess that I cannot think of any other example that is of

genuine interest,

One direction for further investigation in the literature is the

tightening of requirements on necessary and sufficient conditions, For

instance, by requiring that the measure assigned to the finite structure

be unique up to some classical group of transformations like the group

of linear or similarity transformations, we could increase the require­

ments for necessary and sufficient conditions and in this way find simpler

solutions in some cases, I cannot, however, at this time report any non­

trivial positive results in this direction,

3, Algebraic Theory of Error

The easiest place to begin looking at the problem of error is in

simple orderings, We may ask how the problem of error can be introduced

without going to probabilistic considerations, In other words, what can

we say at the algebraic level about a theory of error combined with the

theory of order? It is pleasant to report that in this case a simple

solution is at hand, Surprisingly, the idea was not introduced in an

explicit way much earlier in the literature, The concept of a semiorder,

combining the ideas of error and order, was introduced first by Luce

(1956), and the axioms were simplified later by Scott and me (1958),

Axioms are stated for binary structures U[ = < A, P >, where the

intended interpretation of the relation P is that of strict precedence

or strict preference, 'rhe axioms are just the following three for any

x, y, u, and v in A:

9

Axiom L Not xPx---Axiom 2. If xPy and uPv then either xPv or uPy---Axiom 3. If xPy and uPv then either wPy or uPw---

The intuitive idea of the axioms in the representation is that objects

not in the relation of strict precedence fall below a threshold of dis-

crimination. The theorem that can be proved is that in the finite case

the relation P may be represented as follows:

(2) xPy if and only if f(x) > f(y) + E, with E > 0 •

In reference to our earlier discussion, it also is pleasing to report

that these three axioms are necessary and sufficient for the kind of

representation just shown when' the set A is finite.

A natural part of the algebraic theory of error is to disregard

considerations of order and to consider only the relation between objects

nab discriminable, that is, objects that lie within the threshold of

discrimination. Let I stand for such a relation of indiscriminability.

It is apparent that the relation I should be reflexive and symmetric--

the following two axioms are satisfied for any two objects x and y

in the basic set A of the structure:

~L xIx

Axiom 2. If xly then ylx •

It is also clear what our intended representation for the set I is.

We want to find a representation of the following sort:

(3) xly if and only if /f(x) - f(Y)/ < E with E>O

One would expect that similar simple necessary and sufficient conditions

for the relation I could be found as in the case of the relation of

10

strict precedence given above, Once again, however, the answer is

negative, This recently was proved by Fred Roberts (1968) in his

Stanford dissertation, The result is similar to the one obtained

earlier for difference structures, No finite set of elementary open

sentences can characterize the necessary and sufficient conditions to

obtain the representation re~uired by (3), In the case of the indis-

criminability relation, the heart of the proof lies in the fact that

indiscriminability cycles of ever greater size must be excluded, Let

a connecting line segment indicate that the objects are indiscriminable,

We must exclude cycles of arbitrary size as shown in Figure I, Excluding

Insert Figure 1 about here

such cycles means that in examining specific data on indiscriminability,

we cannot check on the existence of a representation function just by

looking, as we might like to, at pairs or triples or ~uadruples of

objects, We must check n-tuples of arbitrary size to make sure that

a representation exists,

Again, as in Scott's (1964) results for difference structures,

Roberts (1968) does give an infinite set of necessary and sufficient

conditions for representability of relations of indiscriminability on

finite sets, The most important axiom schema is the one that excludes

the countable list of cycles of the kind described above,

The axiomatic situation with respect to algebraic theory of error

for more complicated measurement structures is not yet entirely satis-

factory, Some relatively complicated necessary and sufficient conditions

for the measurement of subjective probability with a semiorder replacing

11

o o '"Figure 1. Cycles of indiscrimin~bility.

12

the ordinary simple ordering are given in the Stanford dissertation of

Zoltan Domotor (1969), Earlier nonaxiomatic work on semiorders in ex­

tensive measurement can be found in Krantz (1967),

These difficulties of finding a workable formulation cast doubt,

it seems to me, upon the practicability of applying algebraic theories

of error to real data, In fact, I know of no place in the experimental

literature where the ideas I have just described are used in a systematic

way, Probabilistic methods or some other approach--for example, the

kind of thing that can be got from using linear programming techniques-­

has been used in all the studies I know about, It seems to me that an

important problem for measurement theory is to determine at a deeper

level whether there are serious possibilities of actually applying the

algebraic theory of error to real experimental data,

4, Nonalgebraic Theory of Error

Standing apart from fundamental theories of error is a very sub­

stantial applied theory about the analysis of error in measurement, It

is not possible here, in the short space that remains, to review this

literature, but certain parts of it are so close to problems of measure­

ment in the social sciences that I would be remiss not to mention their

connection to the topics I have already dwelt upon, Of the normal family

of general methods, none is more common than that of regression, and the

brief remarks I make here can be restricted to regression models without

any loss, The first thing to note about such models is the absence of

an axiomatic basis in the subject matter of the phenomena being measured,

As Professor Williams rightly emphasized in his lecture earlier in the

Seminar, the appropriate use of regression models is in the exploration

of an area and in the identification of relevant variables. The framework

of regression itself does not provide a natural setting in which to inves­

tigate the properties of the structures generic to the domain. As he also

emphasized, we desire to pass from regression models to structural models

that postulate more about the mutual relationships holding in the given

domain. The widespread use of regression models in every area of the

social sciences bears witness to two things--first, to the superficiality

as yet of much of our theorizing, and second, to the absolute necessity

of taking account of errors in the relationships we study. It typically

is the case in a regression analysis that the error term does not really

refer to errors in the measurement of the variables but rather to the

inability of the interrelationship among the independent variables to

account for variations in the dependent variable.

Still another way to state the matter is this. We can hope to find

underlying structural models that will justify, at least as a first

approximation, the regression models that are so easily applied to the

study of phenomena in almost every domain. I would like to stress the

subtlety of the relation that can exist between structural models and

regression models by mentioning very briefly some of my own work. We

recently have begun to apply probabilistic automata models to the analysis

of performance on arithmetical tasks by young children. These probabilistic

automata constitute structural models, I believe, in almost anyone's sense

of the term. We attempt to give a detailed processing account of steps

the students execute in applying an algorithm for finding a numerical

answer. The probabilistic aspects of the automaton are adjusted to data

to fit the errors made by students. Of course, if students made no errors

l~

[. whatsoever, the experimental study would be trivialized,and we could

represent their behavior by a simple finite deterministic automaton.

Under a natural set of assumptions about sources of errors, we can pass

from the automaton model to actual analysis of the data in terms of a

simple regression model. by taking the logarithm of the probability of

a correct answer 0

In a situation like this, the regression model is no longer one

used simply to explore relevant variables; it is an important statistical

tool in testing a rather elaborate underlying structural model. The

theory of error built into the regression model is then directly useful

in fitting the automaton model to the experimental data.

I realize that I have said only the most superficial things about

nonalgebraic error theory, but it is necessary to bring discussion to

an end at this point.

There is one final thing that I would like to say as an expression

of my own feelings about measurement. As we explore a new area of science,

as we develop new insights into a familiar area, or as we improve our

techniques of measurement, we should use those techniques to identify

important variables and to move from that identification to assumptions

that go beyond the theory of measurement itself. Another way of putting

the matter is this. I think that all of us interested in the theory of

measurement should keep an eye cocked at all times for the more general

conceptual fr:amework within which we are working and try to use the

results of measurement to deepen our understanding of that framework,

and even on occasion, totally to rebuild it. In the social sciences

especially, sDund use of the theory of measurement can contribute as

much to mattera of general theory construction as to the improvement of

our empirical methods of investigation and data analysis.

References

Domotor, Zo Probabilistic relational structures and their applications.

Technical Report No. 144, May 14, 1969, Stanford University,

Institute for Mathematical Studies in the Social Sciences.

Krantz, Do H•. Extensive measurement in semiorders. Philosophy of

Science, 1967, 34, 348-3620

Luce, R. D. Semiorders and a theory of utility discrimination.

Econometrica, 1956, 24, 178-1910

Roberts, Fo S. Representations of indifference relations. January 18,

1968, Stanford University, Department of Mathematics, National

Science Foundation Grants GP-4784 and GP-7655.

Scott, D. Measurement structures and linear inequalities. Journal of

Mathematical Psychology, 1964, 1, 233-247.

Scott, D., and Suppes, Po Foundational aspects of theories of measuremento

Journal of Symbolic Logic, 1958, 28, 113-128.

Suppes, P. Axioms for relativistic kinematics with or without parity.

In L. Henkin, P. Suppes, & Ao Tarski (Eds.), The axiomatic methodo

Amsterdam: North-Holland,::L959. Pp. 291-3070

Titiev, R. J. Some model-theoretic results in measurement theory.

Technical Report No. 146, May 22, 1969, Stanford University,

Institute for Mathematical Studies in the Social Scienceso

Zeeman, E. Co Causality implies the Lorentz groupo Journal of

Mathematical Physics, 1964, ~, 490-4930